Graded category

In mathematics, if $𝒜$ is a category, then a $𝒜$-graded category is a category $𝒞$ together with a functor $F:𝒞\to 𝒜$.

Monoids and groups can be thought of as categories with a single object. A monoid-graded or group-graded category is therefore one in which to each morphism is attached an element of a given monoid (resp. group), its grade. This must be compatible with composition, in the sense that compositions have the product grade.

Definition

There are various different definitions of a graded category, up to the most abstract one given above. A more concrete definition of a graded abelian category is as follows:

Let $𝒞$ be an abelian category and $G$ a monoid. Let $𝒮=\{S_g:g\in G\}$ be a set of functors from $𝒞$ to itself. If

• $S_1$ is the identity functor on $𝒞$,
• $S_g{S}_h=S_{gh}$ for all $g,h\in G$ and
• $S_g$ is a full and faithful functor for every $g\in G$

we say that $(𝒞,𝒮)$ is a $G$-graded category.

See also

• Differential graded category
• Graded (mathematics)
• Graded algebra
• Slice category